One of the names most readily associated with one of the greatest revolutions in modern mathematics, the theory of sheaves, is that of Jean Leray. The story of his development of this magnificent tool that revolutionized algebraic geometry and various allied fields (or even not so closely allied: I first encountered sheaves in the context of a graduate course complex variables) is rather dramatic: Leray, as a French officer in World War II, had been interred in a Nazi POW camp and, fearing that if they knew of his expertise in applicable mathematics he would be forced to do war-related work, presented himself as an algebraic topologist. In due course Oflag XVIIA, the POW camp in question, housing a number of French scientist and academics, developed an ersatz university (founded by Leray) in which Leray lectured; around 500 degrees were awarded there, all later officially recognized. There is quite a bit of information available on-line about this remarkable story, including Haynes Miller’s Leray in Oflag XVIIA.

In this horrific environment Leray produced works of note, including the first formulation of sheaf theory, i.e. sheaf cohomology, and the technique of spectral sequences. The papers were of course only published after the war, as the chronology of articles printed in the book under review demonstrates. They appear in Part I of the three-volume set of oeuvres under review, preceded by a fabulous introduction by Armand Borel which describes the marvelous breadth and depth of Leray’s work in topology. Volume I, subtitled “Topology and Fixed Point Theorems,” contains articles ranging from 1934 to 1972, indicating the span of Leray’s involvement in the field.

Volume II’s contents demonstrate the wisdom Leray showed in hiding his other expertise from his Nazi captors: it is subtitled “Fluid Dynamics and Real Partial Differential Equations,” and is chock-full of this kind of hard-core analysis. This volume is introduced by Peter Lax and includes Leray’s 1933 dissertation on non-linear integral equations and applications in hydrodynamics. The first article presented in the volume, however, is a 1972 speech by Leray on mathematics and its applications, making for very interesting reading, not least because Leray allows himself to philosophize about the role of mathematics in the world of the day.

Finally, Volume III, introduced by Guennadi Henkin, deals with several complex variables, and we read from Henkin’s Introduction that

… On the one hand, for Leray problems of complex analysis have never been the main purpose of his research. On the other hand, in his works on differential equations and mathematics physics Leray systematically and with great success used not only various methods of algebraic topology and partial differential equations but also methods of contemporary complex analysis … Without exaggeration one can say that during the fifties-sixties the ideas of Leray twice radically changed the direction of contemporary complex analysis.

Indeed, Jean Leray was a versatile scholar of the highest order.

One thing that is not included in this marvelous three-volume set of Leray’s selected works, is of course his book, Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index, which is revolutionary in its own right. The Maslov index was christened by V. I. Arnol’d in his fundamental paper, “Characteristic class entering in quantization conditions,” and it was Arnol’d who, as the book’s dust-jacket indicates,

asked Jean Leray — one of this century’s most prominent masters of both pure and applied mathematics — what the implications of Maslov’s procedure might be. Lagrangian Analysis and Quantum Mechanics represents Leray’s answer to Arnol’d’s question.

And this is a fitting epitaph with which to end this review: Leray was indeed one of the great masters of both pure and applied mathematics, as these three volumes of his oeuvres amply demonstrate. A closing caveat: all but two of the articles (one, dating to 1983, dealing with the Maslov index and Planck’s constant, the other, dating to 1972, dealing with fixed point theorems and Lefschetz’ number) are in French. C’est la vie.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.